Q5. Find the point that divides the segment from P(1, −2, 3) to Q(3, 4, −5) externally in the ratio 1 : 3.
((1 × 3 − 3 × 1)/(1 − 3), (1 × 4 − 3 × (−2))/(1 − 3), (1 × (−5) − 3 × 3)/(1 − 3)) = (0, −5, 7).
Q6. Show that A(2, −3, 4), B(−1, 2, 1) and C(0, 1/3, 2) are collinear by finding the ratio in which C divides AB.
Let C divide AB in the ratio k : 1. From x: (−k + 2)/(k + 1) = 0, so k = 2. Check y: (2 × 2 + (−3))/3 = 1/3, and z: (2 × 1 + 4)/3 = 2. One ratio fits all three coordinates, so C lies on AB and divides it in the ratio 2 : 1.
Q7. In what ratio does the ZX-plane divide the segment from P(−3, 4, −8) to Q(5, −6, 4)? Find the point where it cuts it.
k = −y1/y2 = −4/(−6) = 2/3, so the ratio is 2 : 3, internally. The point is ((2 × 5 + 3 × (−3))/5, 0, (2 × 4 + 3 × (−8))/5) = (1/5, 0, −16/5).
Q8. Three vertices of a parallelogram ABCD are A(1, 2, 3), B(−1, −2, −1) and C(2, 3, 2). Find D.
D = A + C − B = (1 + 2 + 1, 2 + 3 + 2, 3 + 2 + 1) = (4, 7, 6). These are the vertices of NCERT Example 7, page 213.
Q9. Find the centroid of the tetrahedron with vertices (0, 0, 0), (4, 0, 0), (0, 8, 0) and (0, 0, 12).
The average of the four vertices: (4/4, 8/4, 12/4) = (1, 2, 3).
Q10. The mid-point of AB is (2, −1, 3) and A is (5, 1, −2). Find B.
B = 2 × (mid-point) − A = (4 − 5, −2 − 1, 6 + 2) = (−1, −3, 8).
Q11. Find the two points that divide the segment from (4, −1, 2) to (−2, 5, 8) into three equal parts.
The ratios are 1 : 2 and 2 : 1: ((−2 + 8)/3, (5 − 2)/3, (8 + 4)/3) = (2, 1, 4) and ((−4 + 4)/3, (10 − 1)/3, (16 + 2)/3) = (0, 3, 6).
Q12. Find the ratio in which R(5, 4, −6) divides the segment from P(3, 2, −4) to Q(9, 8, −10).
With the ratio k : 1, (9k + 3)/(k + 1) = 5 gives k = 1/2. The y- and z-coordinates agree: (8 × 1/2 + 2)/(3/2) = 4 and (−10 × 1/2 − 4)/(3/2) = −6. The ratio is 1 : 2, internally.